If alpha ,beta ,gamma are the roots of | vedclass

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Question

$\alpha \,is\,a\,root\, \Rightarrow \,{\alpha ^3} = \,\alpha  + 2 $
 $  \Rightarrow {\alpha ^5} = {\alpha ^3} + 2{\alpha ^2} = \alpha&

Vedclass · Accepted Answer

$10$

Vedclass · Answer

$\alpha \,is\,a\,root\, \Rightarrow \,{\alpha ^3} = \,\alpha  + 2 $
 $  \Rightarrow {\alpha ^5} = {\alpha ^3} + 2{\alpha ^2} = \alpha  + 2 + 2{\alpha ^2} $
$  	herefore {\alpha ^5} + {\beta ^5} + {\gamma ^5} = 2({\alpha ^2} + {\beta ^2} + {\gamma ^2}) + ({\alpha ^{}} + \beta  + \gamma ) + 6 $
$   = 2(\sum \alpha  - 2\sum \alpha \beta ) + \sum \alpha  + 6$
$   = 2(0 - 2( - 1)) + 0 + 6 = 10 $

If $\alpha ,\beta ,\gamma$ are the roots of $x^3 - x - 2 = 0$, then the value of $\alpha^5 + \beta^5 + \gamma^5$ is-

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